CBSE · Class 12 · Mathematics · Chapter 9

Differential Equations — Formula Sheet

Board Formulas

16 formulas

1.Order of a Differential Equation
$Order = order of the highest derivative present$
Look only at which derivatives appear, not at their powers. Order is always a positive integer.
2.Degree of a Differential Equation★
$x (\frac{d ^{2} y}{d x ^{2}})^{3} + (\frac{d y}{d x})^{4} - y = 0 \Rightarrow order 2, degree 3$
Degree is the highest power of the HIGHEST-ORDER derivative, once the equation is a polynomial in its derivatives. The 4th power of dy/dx does not count here.
3.When Degree is Not Defined
$\frac{d ^{2} y}{d x ^{2}} + sin (\frac{d y}{d x}) = 0 \Rightarrow degree not defined$
A derivative inside sin, cos, e^( ), log and so on means the equation is not a polynomial in its derivatives, so its degree is not defined.
4.General and Particular Solutions
$order n \Rightarrow general solution has n arbitrary constants$
A particular solution has no arbitrary constants. It is obtained by fixing the constants from given conditions.
5.Particular Solution from a Condition
$y (x_{0}) = y_{0} \Rightarrow substitute in the general solution to find C$
Use the condition only after the general solution is complete. Substituting early often loses terms.
6.Variables Separable★
$\frac{d y}{d x} = g (x) h (y) \Rightarrow \int \frac{d y}{h ( y )} = \int g (x) d x + C$
$g (x)$ : Factor depending on x only · $h (y)$ : Factor depending on y only · $C$ : Arbitrary constant
Get every y-term with dy and every x-term with dx, then integrate both sides. Requires h(y) ≠ 0.
7.Homogeneous Function of Degree n
$F (λ x, λ y) = λ^{n} F (x, y)$
$λ$ : Any non-zero constant · $n$ : Degree of homogeneity
Replace x by λx and y by λy. If λ factors out as λⁿ, F is homogeneous of degree n. Equivalently, F(x, y) = xⁿ g(y/x).
8.Homogeneous Differential Equation
$\frac{d y}{d x} = F (x, y), F homogeneous of degree 0 \Leftrightarrow \frac{d y}{d x} = g (\frac{y}{x})$
To test, show F(λx, λy) = λ⁰F(x, y) = F(x, y). Write this test in your answer; it is part of a complete solution.
9.Substitution y = vx★
$y = v x \Rightarrow \frac{d y}{d x} = v + x \frac{d v}{d x}$
This turns a homogeneous equation into a separable equation in v and x. Do not forget the 'v +' term.
10.Homogeneous Equation after Substitution
$v + x \frac{d v}{d x} = g (v) \Rightarrow \int \frac{d v}{g ( v ) - v} = \int \frac{d x}{x} + C$
Integrate, then replace v by y/x to get the general solution.
11.Substitution x = vy
$\frac{d x}{d y} = h (\frac{x}{y}) : x = v y \Rightarrow \frac{d x}{d y} = v + y \frac{d v}{d y}$
Use this when the equation is more naturally written as dx/dy, for example when terms like e^(x/y) appear.
12.Linear Differential Equation
$\frac{d y}{d x} + P y = Q, P, Q constants or functions of x only$
$P$ : Coefficient of y (function of x or constant) · $Q$ : Right-hand side (function of x or constant)
y and dy/dx appear only to the first power and are never multiplied together. Bring the equation to exactly this form first.
13.Integrating Factor★
$I.F. = e^{\int P d x}$
Leave out the constant when integrating P. Simplify using e^(log f) = f. Here log means the natural logarithm, as in NCERT.
14.Solution of a Linear Equation★
$y \cdot (I.F.) = \int Q \cdot (I.F.) d x + C$
After multiplying by the I.F., the left side is exactly d/dx[y × I.F.]. Integrate the right side (often by parts), then divide by the I.F. if y is needed explicitly.
15.Linear Equation in x
$\frac{d x}{d y} + P_{1} x = Q_{1} : I.F. = e^{\int P_{1} d y}, x \cdot (I.F.) = \int Q_{1} \cdot (I.F.) d y + C$
Use this when the equation is linear in x but not in y, for example y dx − (x + 2y²) dy = 0. Here P₁ and Q₁ are functions of y only.
16.Useful Simplifications of the I.F.
$e^{l o g x} = x, e^{- l o g x} = \frac{1}{x}, e^{k l o g x} = x^{k} (x > 0)$
Also e^(log sec x) = sec x and e^(−log cos x) = sec x for values where these functions are positive.